rational expressions chapter 5. 5-1 quotients of monomials
TRANSCRIPT
RATIONAL EXPRESSIONS
Chapter 5
5-1 Quotients of Monomials
Multiplication Rule for FractionsLet p, q, r, and s be real numbers with q ≠ 0 and s ≠ 0. Then
p •r = pr q s qs
Rule for Simplifying FractionsLet p, q,and r be real numbers with q ≠ 0. Then
pr = p qr q
EXAMPLES
Simplify:30
40Find the GCF of the numerator and denominator.
EXAMPLES
GCF = 1030 = 10 • 3
40 10 • 4
= 3/4
EXAMPLES
Simplify:9xy3
15x2y2
Find the GCF of the numerator and denominator.
EXAMPLES
GCF = 3xy2
9x3 = 3y • 3xy2
15x2y2 5x • 3xy2
= 3y/5x
LAWS of EXPONENTS
Let m and n be positive integers and a and b be real numbers, with a ≠ 0 and b≠0 when they are divisors. Then:
Product of Powers
•am • an = am + n
•x3 • x5 = x8
•(3n2)(4n4) = 12n6
Power of a Power
•(am)n = amn
•(x3)5 = x15
Power of a Product
•(ab)m = ambm
•(3n2)3 = 33n6
Quotient of Powers
If m > n, thenam an = am-n
x5 x2 = x3
(22n6)/(2n4) = 11n2
Quotient of PowersIf m < n, thenam an = 1/a|m-n|
x5 x7 = 1/x2
(22n3)/(2n9) = 11 n6
QUOTIENTS of MONOMIALS are SIMPLIFIED
When:• The integral coefficients
are prime (no common factor except 1 and -1);
• Each base appears only once; and
• There are no “powers of powers”
EXAMPLES
Simplify24s4t3
32s5
EXAMPLES
Answer3t3
4s
EXAMPLES
Simplify-12p3q
4p2q2
EXAMPLES
Answer-3p
q
5-2 Zero and Negative Exponents
Definitions
If n is a positive integer and a ≠ 0
a0= 1 a-n = 1
an
00 is not defined
Definitions
If n is a positive integer and a ≠ 0
a-n = 1/an
EXAMPLES
Simplify:-2-1a0b-3
EXAMPLES
Answer: -1
2b3
5-3 Scientific Notation and
Significant Digits
DefinitionsIn scientific notation, a number is expressed in the form m x 10n where
1 ≤ m < 10 and n is an integer
5-4 Rational Algebraic
Expressions
DefinitionsRational Expression – is one that can be expressed as a quotient of polynomials, and is in simplified form when its GCF is 1.
EXAMPLES
Simplify:x2 - 2x
x2 – 4
Factor
EXAMPLESAnswer:
x (x – 2) (x + 2)(x - 2)
= xx + 2
DefinitionsA function that is defined by a simplified rational expression in one variable is called a rational function.
EXAMPLES
Find the domain of the function and its zeros.
f(t) = t2 – 9 t2 – 9t FACTOR
EXAMPLESAnswer:
(t + 3) (t – 3) t(t – 9)Domain of t = {Real numbers except 0 and 9}
Zeros are 3 and -3
Graphing Rational Functions
1.Plot points for (x,y) for the rational function.
2. Determine the asymptotes for the function.
3.Graph the asymptotes using dashed lines. Connect the points using a smooth curve.
Definitions
Asymptotes- lines approached by rational functions without intersecting those lines.
5-5 Products and Quotients of Rational
Expressions
Division Rule for Fractions
Let p, q, r , and s be real numbers with q ≠ 0, r ≠ 0, and s ≠ 0.
Then p r = p • s q s q r
EXAMPLES
Simplify.
6xy 3y a2 a3x
EXAMPLESAnswer:
= 6xy •a3x a2 3y
= 2ax2
5-6 Sums and Differences of
Rational Expressions
Addition/Subtraction Rules for Fractions
1.Find the LCD of the fractions.
2. Express each fraction as an equivalent fraction with the LCD and denominator
3.Add or subtract the numerators and then simplify the result
EXAMPLESSimplify:
= _1 – _1 + _3_ 6a2 2ab 8b2
4b2 – 12ab + 9a2
= 24a2b2
= (2b – 3a)2
24a2b2
5-7 Complex Fractions
DefinitionComplex fraction – a
fraction which has one or more fractions or powers with negative exponents in its numerator or denominator or both
Simplifying Complex Fractions
1.Simplify the numerator and denominator separately; then divide, or
Simplifying Complex Fractions
2.Multiply the numerator and denominator by the LCD of all the fractions appearing in the numerator and denominator.
EXAMPLES
Simplify: (z – 1/z) (1 –
1/z)
Use Both Methods
5-8 Fractional Coefficients
EXAMPLES
Solve:
x2 = 2x + 1 2 15 10
5-9 Fractional Equations
DefinitionFractional equation
– an equation in which a variable occurs in a denominator.
DefinitionExtraneous root – a
root of the transformed equation but not a root of the original equation.
END